Matrix extension and biorthogonal multiwavelet construction

Abstract

Suppose that P( z) and P ̃ (z) are two r × n matrices over the Laurent polynomial ring R[ z], where r < n, which satisfy the identity P(z) P ̃ (z)∗ = I r on the unit circle T . We develop an algorithm that produces two n × n matrices Q( z) and Q ̃ (z) over R[ z], satisfying the identity Q(z) Q ̃ (z)∗ = I n on T such that the submatrices formed by the first r rows of Q( z) and Q ̃ (z) are P( z) and P ̃ (z) respectively. Our algorithm is used to construct compactly supported biorthogonal multiwavelets from multiresolutions generated by univariate compactly supported biorthogonal scaling functions with an arbitrary dilation parameter m ∈ Z, where m >1.